Abstract

In this paper, we present the theoretical research results of certain characteristics of the generalized hyperspherical function with two degrees of freedom as independent dimensions. Here, we primarily give the answers to the quantification of dimensional potentials (fluxes) of this function in the domain of natural numbers. In addition, we also give the solutions to continual fluxes of separate contour hyperspherical (HS) functions. The symbolical evaluation and numerical verification of the values of series and integrals are realized using MathCAD Professional and Mathematica.

MSC 2010: 51M04; 33E99.

Keywords

special functionhyperspheredimension flux

1. Introduction

The hypersphere function is a hypothetical function related to multi-dimensional space (see [1–3]). The most important aspect of this function is its connection to all functions that describe the properties of spherical entities: points, diameter, circumference, circle, surface, and volume of a sphere. The second property is the generalization of these functions from discrete to continuous. It belongs to the group of special functions, so its testing is being performed on the basis of known functions such as gamma (Γ), psi (ψ), and the like, so that its generalized, explicit form is the following [4].

Definition 1.1.The hyperspherical function [5]with two degrees of freedom k and n is defined as

HS(k,n,r)=2πkrk+n-3Γ(k)Γ(k+n-2)Γk2k,n∈Z,r∈N,

(1.1)

where Γ(z) is the gamma function.

Using the fundamental properties of the gamma function, we advance from the domain of the natural values analytically to the set of real values for which we form the conditions for both its graphical interpretation and a more concise mathematical analysis. It is developed on the basis of two degrees of freedom k and n as vector dimensions, in addition to radius r, as an implied degree of freedom for every hypersphere. The dominant theorem is the one that relates to the recurrent property of this function [6].

It implies that the vectors on the left (n = 2, 1, 0, -1, -2,...) of the matrix M[HS]kxn (1.3) are obtained on the base of the reverent vector (n = 3) deduction, and the vectors on the right (n = 4, 5, 6, 7, 8,...) on the base of integrals by radius r [7].

2. Dimensional potentials--the fluxes of HS function

2.1. Vertical dimensional flux of hypersphere function

Definition 2.1. The discrete dimensional potential or the hypersphere function flux is the sum of all separate functions in the (sub)matrix of this function that we expand for integer or real degrees of freedom (1.3).

Formally, this flux can be quantified by twofold series that covers this area of the HS function. The first phase is to define the value of infinite series of functions classified in columns (vectors) of the submatrix M[HS]kxn (k, n∈N). This is also the definition of vertical dimensional fluxes of HS function. The first value to be calculated relates to the fourth columns (n = 3) of this submatrix (1.3). From this, we obtain

Consequently, Freden's result [27] can be presented in the form of series with even (k = 2b) and odd members (k = 2b+1, where b∈N). In this sense we get

eπr2=∑b=0∞πbr2bb!andeπr2erf(rπ)=∑b=0∞πb(2r)2b+1b!(2b+1)!.

On the base of Freden's solution (2.1), as a starting point and applying the recurrent relations (1.2), we get values for lower degrees of freedom (n < 3). We connect the sphere hypervolume (n = 3) with its hypersurface (n = 2). In this sense a new vector flux follows

∂∂r∑k=0∞HS(k,3,r)=∑k=0∞HS(k,2,r),so then is∑k=0∞HS(k,2,r)=21+πreπr2erfc(-rπ).

For hypersphere (n = 1), a series is obtained in view of the previous, therefore

∂∂r∑k=0∞HS(k,2,r)=∑k=0∞HS(k,1,r)=2π2r+eπr2(1+2πr2)erfc(-rπ)

For n = 0, the series value is found on the basic of deducing, so it follows that

∑k=0∞HS(k,0,r)=8π2reπr2(32+πr2)erfc(-rπ)+r2+π-1.

For degrees of freedom higher than n = 3, series are found by inverse operations, i.e., by recurrent relation in view of integrating along radius r. Consequently,

∑k=0∞HS(k,n+1,r)=∫0r∑k=0∞HS(k,n,r)dr,

so for the fourth dimension the following integral form is applied

∑k=0∞HS(k,4,r)=∫0r∑k=0∞HS(k,3,r)dr=∫0reπr21+erf(rπ)dr.

This property refers also to the complementary dichotomous hyperspherical series. In that case, we have for even members

∑k=0,2,4,…∞HS(k,4,r)=∫0r∑k=0,2,4,⋯∞HS(k,3,r)dr.

So for the fourth dimension an integral form with imaginary error function is obained using erf(z) = -ierf(iz).

∑k=0,2,4,…∞HS(k,4,r)=∫0reπr2dr=-i2erf(r-π)=erfi(rπ)2.

After the partial integration using ∫udv=uv-∫vdu we obtain the integral for "odd" series

where u=erf(rπ) and dv=eπr2dr. The addend with the integral on the right side is analytically solvable and it amounts to, e.g., the series with "b" members (or as a series with the incomplete gamma function). In that sense, the value of the series with even numbers is

2.2. Integral solvability on the base of the incomplete gamma function

The flux for n = 3 is the easiest one to solve, and it represents the base for calculating fluxes of higher degrees of freedom (n > 0), through integration of previously obtained results. In that sense, this procedure is possible by using the series where the incomplete gamma function. The second integral (2.2) is reduced to known terms, and one among them is [28]

∫ebz2erf(az)dz=1bπ∑k=0∞a2k+1Γ(k+1,-bz2)bk(2k+1)k!+C.

Integral in its definite form is expressed as

∫0reπr21+erf(rπ)dr=erfi(rπ)2+1π∑k=0∞Γ(k+1,-πr2)-Γ(k+1,0)(1+2k)k!.

While the incomplete gamma function is in general case equal to Γ(a,z)=∫z∞ta-1e-tdt[29]. It is obvious that the two obtained series with odd members are equivalent to the following

1π∑k=0∞Γ(k+1,-πr2)-Γ(k+1)(1+2k)k!≡∑b=0∞21+2bπbr2(1+b)b!(2+2b)!.

Defining the series for n = 5 is realized by integrating the expressions

∑k=0∞HS(k,5,r)=∫0rerfi(rπ)2dr+∫0r∑k=1,3,…∞2kπk-1rk+1(1+k)!k-12!dr.

(2.3)

The first integral on the right side of (2.3) is solved on the bases of the known equality ∫erfi(az)=z⋅erfi(az)-e(az)2aπ, where az=rπ. Thus, the integral is obtained as

∫0rerfi(rπ)2dr=12r⋅erfi(rπ)-eπr2π0r=1-eπr2+πrerfi(rπ)2π.

However, a simpler way is to use both even and odd dichotomous series. Now, we obtain

Note: Integration, similar as in the previous cases, is applied with certain conditions, so we have, e.g., ∫0rr2b+3dr=0, if -1 < 2Re(b) + 3 < -1^b ≠ -1^r = ∞. If the conditions are not met, this integral is indefinite. Some values of these discrete and continuous fluxes (for r = 1) are given in Table 1.

Table 1

Values of discrete and continuous fluxes.

Degree of freedom (n)

∑k=0∞HS(k,n,1)≈

∫0∞HS(k,n,1)dk≈

0

16962.1740457

16962.3520362

1

2117.56926532

2117.48007283

2

291.022289825

291.104223905

3

45.9993260894

45.5712471365

4

8.71952109668

8.20993584833

5

1.87596579993

1.60128605246

6

0.40326040109

0.30739217922

7

0.07910676340

0.05435115208

8

0.01367865325

0.00860415949

9

0.00207449183

0.00121196056

10

0.00027764247

0.00015240808

11

0.00003309744

0.00001722662

12

0.00000354778

0.00000176333

⋮

⋮

⋮

∞

limn→∞∑k=0∞HS(k,n,r)=0

limn→∞∫0∞HS(k,n,r)dk=0

Σn

19427.858848843922

-

Suppose that the values of the vector fluxes decline with the increase of the degree of freedom n. The dimensional fluxes can be studied as well for the complex part. So, for example, with the recurrence we get the series values for the negative degree of freedom n = -2, as [30].

∑k=0∞HS(k,-2,r)=8π3reπr2erfc(-rπ)[15+4πr2(5+πr2)]+2r2(9+2πr2)+8π-1.

2.3. Fluxes on the base of hypersphere matrix series

The discrete dimensional fluxes can be calculated as well "horizontally", i.e., by adding function values along the M[HS]kxn submatrix series. For example, by expanding the series for k = 3, the flux would contain the following members

∑n=0∞HS(3,n,r)=8π+8πr+4πr2+43πr3+13πr4+115πr5+…+ε8πrnΓ(n+1).

Some values of discrete and continual fluxes, (for r = 1), are given in Table 2.

Table 2

Values of discrete and continual fluxes

Degree of freedom (k)

∑n=0∞HS(k,n,r)=

∑n=0∞HS(k,n,1)≈

∫0+∞HS(k,n,1)dn≈

0

er

2.71828182846

2.89982256317

1

2er

5.43656365692

5.24809906025

2

2πer

17.0794684453

17.6417407306

3

8πer

68.3178737814

56.964225268

4

12π2(er-1)

203.505142758

139.918441638

5

64π2(er-r-1)

453.706079704

271.32230045

6

60π3[2er-(r+1)2-1]

812.172812098

437.960809928

7

∑n=0∞768π3rn+4Γ(n+5)

1229.10258235

611.722905550

8

∑n=0∞1680π4rn+5Γ(n+6)

1628.04409715

759.633692941

9

∑n=0∞12288π4rn+6Γ(n+7)

1933.28876014

855.051695653

10

∑n=0∞30240π5rn+7Γ(n+8)

2093.93742907

884.975895298

11

∑n=0∞245760π5rn+8Γ(n+9)

2095.29352414

851.441651487

12

∑n=0∞665280π6rn+9Γ(n+10)

1956.27052708

768.011397877

13

∑n=0∞5898240π6rn+10Γ(n+11)

1717.51550066

653.938458847

⋮

⋮

⋮

⋮

50

∑n=0∞HS(50,n,r)

0.00000002078

0.00000000526

⋮

⋮

⋮

⋮

∞

limk→∞∑n=0∞HS(k,n,r)=0

0

limk→∞∫0∞HS(k,n,r)dn=0

Σk

19427.858848843922

-

2.4. Some continuous fluxes

The distribution trend of vector fluxes is increasing, followed by asymptotic decrease with linear growth of degree of freedom n. From the standpoint of functional analysis, the most interesting series of the matrix M[HS]k, n is the one that relates to the degrees of freedom k = 2 and k = 3. The first series includes the known functions for the circumference (2πr) and the surface of circle (πr2). The members of the second series are the surface functions (4πr2) and sphere volume (43πr3). The same series are interesting as well for continuous fluxes. The continuous natural flux for the hypersphere surface is analyzed on the base of integrals, instead of series. This integral is specific, because its subintegral function is the reciprocal gamma function. Its value, as it is known, is equal to the value of Fransen-Robinson constant [31]

F=∫0∞1Γ(x)dx=e+∫0∞e-nπ2+ln2ndn≈2.8077702420285.

The integral value of the flux in question is now ∫0∞HS(2,n,1)dn=∑n=0∞HS(2,n,1)+∫0∞2πe-nπ2+ln2ndn or, namely, for unit radius

∫0∞2πΓ(n)dn=2πe+∫0∞e-nπ2+ln2ndn=2πF≈17.641741.

(2.4)

Regarding the continuous dimension n, a more general dimensional volume hypersphere flux follows on the base of Ramanujan-Hardy's integral [26].

∫0∞yxΓ(x+1)dx=ex-∫0∞e-xzx(π2+ln2x)dx.

Ramanujan defined this integral and Hardy "deepened" it analytically. In that sense, the previous expression can be applied on flux calculation, as

∫0∞8πrnΓ(n+1)dn=8πer-∫0∞e-nrn(π2+ln2ndn.

The integral can be defined as the difference of series and integral with the value (for r = 1),

∫0∞8πrnΓ(n+1)dn=∑n=0∞8πrnΓ(n+1)-∫0∞8πe-nrn(π2+ln2n)dnr=1≈56.96423.

(2.5)

2.5. Vector flux series

Total dimensional flux of the degree of freedom in the domain of natural numbers is obtained as the result of twofold amount by which integer values of hyperspherical function HS(k,n,r), (k, n, r ≥ 0) are respected. This twofold series has to be convergent, and this property is in the function of hypersphere radius. As usual, calculating of total discrete flux is being performed with its unit value and the convergence is in that case provided, taking into consideration that the unit series on that condition are convergent. The flux can be considered also for each column M[HS]k, n of the matrix, separately. So, we have for the n th column (denoted by < n >), the flux in the following form

ΦHS<n>(k,n,r)=∑k=0∞HS(k,n,r).

2.6. Orthogonal dimensional flux

These fluxes are all columns or M[HS]k, n. As the number of columns, respectively, series, is infinite, we introduce the following definition for the total flux.

Definition 2.1.The dimensional flux of the functional matrix with two degrees of freedom k and n is defined as a double series

2.7. The application of the recurring operators at defining diagonal dimension fluxes

In the previous analysis, the defining of the dimensional fluxes of the HS matrix was performed on the basis of addition of the HS function values on the columns, in regard to the series of the HS matrix. The more detailed analysis would be very large scale, including the exponential function, error functions erf(z), erf(z), the incomplete gamma function Γ(a, z), etc. When we use the idea of the transition operators from the reference function into the defining HS function in the functional hyperspherical matrix, we can also establish the values of the dimensional fluxes on the diagonals (Figure 3), whose sum would present the overall flux for the matrix where the degree of freedom is in the domain of natural numbers, i.e., k, n∈N. Such matrix contains an infinite number of elements. For the reference functions, we take HS functions on the positions of the first series of the matrix, and they are the so-called zero HS functions: HS(0,0,r), HS(0,1,r),...,HS(0,n,r),... The defining functions are placed according to the "gradual" law of growth (+Δk) and decline (-Δn).

Figure 3

(a) The representative submatrix. Figure 3. (b) The addition principle of the matrix on diagonal.

Definition 2.2. The flux operator of the series ϑ(Δk, Δn, 0) is defined by the quotient [4]

Also as the absolute values of the increments are equal and unique, that is |Δk| = |-Δn| = 1, a new joint argument u (Δk = Δn = u) is assigned to them. In addition to the starting value of the k th degree of freedom is k = 0, the operator theta becomes (2.4)

θ(u,-u,0)=2πuΓ(u)Γ(u/2).

The assigning function is now being calculated as

HS(u,n-u,r)=θ(u,-u,0)⋅H(0,n,r)=2πurn-3Γ(u)Γ(n-2)Γ(u/2).

The dimensional flux on the diagonal presents the sum of its individual members. So, for the first diagonal (denoted by < 0 >) the flux is equal to

Π<0>(k,n,r)=HS(0,0,r)=0,

for the second, we have

Π<1>(k,n,r)=HS(0,1,r)+HS(1,0,r)=0

for the third

Π<2>(k,n,r)=HS(0,2,r)+HS(1,1,r)+HS(2,0,r)=0,

and for the fourth

Π<3>(k,n,r)=HS(0,3,r)+HS(1,2,r)+HS(2,1,r)+HS(3,0,r)=3+10π.

The flux in the n th diagonal would be calculated in the form of a sum

Π<n>(k,n,r)=∑u=0nHS(u,n-q,r)=2rn-3Γ(n-2)∑u=0nπuΓ(u)Γ(u/2)(n≠2).

(2.7)

The flux for the value n = 2 is calculated on the basis of the function limit value. Respecting that

Γ(u)Γ(u/2)=2u-1πΓu+12,

the expression for the flux of the n th diagonal, after reordering, can get a new form, and it is equivalent to the expression (2.7)

Π<n>(k,n,r)=rn-3Γ(n-2)∑u=0n2uπu-1Γu+12.

So, for the fifth diagonal (n = 4) we get

Π<4>(k,n,r)=∑u=0nHS(u,n-u,r)=r(10π+12π2+3).

For the sixth diagonal (n = 5) it follows that

Π<5>(k,n,r)=∑u=0n=5HS(u,n-u,r)=r22(10π+76π2+3).

As the number of diagonals is infinite, the total flux is formed as the series of all diagonal fluxes

The diagonal flux of the hyperspherical function can be expressed by the series of the general form

ΠHS(v,r)=∑v=0∞avrv.

Here, v is a summing index by which the sequence of the matrix elements from left to right and from above to down along the diagonal is taken into consideration. The polynomial coefficients contain rational numbers and the graded constant π. The first three coefficients are zero, so they are not included in the summation sequence. Its other values (v = 0, 1,...,13) are given in Table 3

Approximately, the double series leads to the solution that is very close to the exact one. Namely, for the unique radius and reducing to ∞ ~ n = 30 the double series of the diagonal flux gets the following structure:

The diagonal dimensional flux is characteristic with coefficients that contain πn constant in the degrees of the series members, in contrast to vertical fluxes with the domination of function errors, where π and e are constants. The horizontal fluxes, as it was presented in (2.6), contain exponential functions. In the meantime, the total flux for the unique radius is convergent and can be calculated with considerably greater value

ΠHS(k,n,1)≈19427.858848843922,

while, e.g., for r = 2 the flux value is substantially greater and its value is obtained as ΠHS≈ 1375905492.377.

3. Conclusion

On the basis of the assumption of the recurrent relations (1.2) that exist within the hyperspherical function (1.1), we can calculate a discrete dimensional flux of this function in the domain of natural integer degrees of freedom. Quantitative flux value depends on the nominated value of the hypersphere radius. Meanwhile, as the function HS(k,n,r) is the function of three variables, its dependence is certainly also a variable k, respectively, n. In this article, we calculated several continual fluxes (2.4) and (2.5), for contour hyperspherical function, on the basis of Ramanujan-Hardy's integral. The dimensional flux calculating with diagonal algorithm is much simpler and faster to perform on a computer, because the total flux is now defined as convergent-graded series and it does not contain special functions as components. In any case its value is identical with the fluxes that are calculated on the base of series, i.e., the HS matrix columns, so there is a numerically verified statement that

ΦHS(k,n,r)=ΩHS(k,n,r)=ΠHS(k,n,r)r=1≈19427.858848843922.

The flux calculating procedure originates from Freden when he defined it in 1993 as the series of hyperspherical functions that refer to the degree of freedom k = 3. In that case, we obtain the solution [20].

∑k=0∞HS(k,3,r)=∑k=0∞πkrkΓk2+1=eπr2erfc(-rπ).

This function belongs to the family of Mittag Feffler's-type functions, which he developed already in the early 20th century [32]. In any case, this solution is initial for solving the other dimensional fluxes, both for hyperspherical and for hypercubic, in other words the hyper-cylindrical function [33]. With continuous flux in the domain k,n∈0,∞¯, the problem is considerably more complex, because for its defining, the double integration (3.1) must be performed. It is supposed that its value is very close to discrete flux that is obtained on the base of twofold series. Total dimensional continual flux (k, n∈N) of the unit hyperspherical function HS(k,n,1) is equal to the value of twofold integral

∫0∞∫0∞2πkrk+n-3Γ(k)Γ(k+n-2)Γk2dkdn,

(3.1)

and its solution can be looked for on numerical bases.

Declarations

Authors' original submitted files for images

Below are the links to the authors’ original submitted files for images.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

DL, the worked to defined the formula for Hipersferičnu function (HS) and the corresponding orthogonal dijalgonalne and fluxes, as a key research contributes to the work. NC, has verified the analitical formulas and numerical basis.

BD, using software packages Mathcad and Mathematica radio program on verification of the obtained expressions and corresponding graphical presentations.

IB, is also working on information support of the overall survey about HS function, and thus this part that is hardest probation, and was presented in this paper. All authors read and approved the final manuscript.

Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.